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For a single qubit in the conventional computational basis, the Bloch sphere maps a pure state to a point on a sphere. The north pole is |0⟩, the south pole is |1⟩, and the equator contains states with equal probabilities of measuring 0 or 1 in that basis. The polar angle θ sets those probabilities; the azimuthal angle φ sets the relative phase. This convention takes θ down from +z and φ around the x-y plane from +x toward +y.
What does a point on the Bloch sphere represent?
It represents the state of one qubit, with the sphere’s axes tied to a chosen basis. For the usual computational basis, a pure state can be written, apart from an irrelevant overall phase, as:
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|ψ⟩ = cos(θ/2)|0⟩ + eiφ sin(θ/2)|1⟩
The corresponding Bloch vector is (sin θ cos φ, sin θ sin φ, cos θ). Thus θ and φ give the direction of the vector: θ is measured down from the positive z axis, while φ is measured in the x-y plane from +x toward +y. These conventions and the state parametrization are described in William A. Girvin’s Introduction to Quantum Information (version dated 2026-03-29) and the Stanford Encyclopedia of Philosophy’s Quantum Computing entry.
What do the poles mean?
With the conventional z-axis assignment, the north pole (θ = 0) is |0⟩, and the south pole (θ = π) is |1⟩. A measurement in the computational, or z, basis returns one of those two outcomes. A point between the poles describes a qubit state, not a third measurement outcome. The Quantum Atlas’s qubit guide explains this pole-and-measurement picture.
What do θ and φ change?
θ sets the z-basis probabilities
The polar angle controls the balance of the two computational-basis outcomes:
P(0) = cos²(θ/2)P(1) = sin²(θ/2)
At the north pole, the state gives outcome 0 with certainty; at the south pole, it gives outcome 1 with certainty. Moving between them changes the probabilities continuously. The half-angle probabilities follow from the qubit parametrization described by the Stanford Encyclopedia of Philosophy and Girvin’s textbook.
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φ sets the relative phase and direction around the sphere
Changing φ rotates the Bloch vector around the z axis and changes the relative phase between |0⟩ and |1⟩. It does not alter the z-basis probabilities, but it does distinguish states and can affect results in other measurement bases. The overall phase multiplying the entire state has no observable effect; the relative phase eiφ does.
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The equator is θ = π/2. There, P(0) = P(1) = 1/2 for a computational-basis measurement. Those equal probabilities do not make all equatorial states identical: changing φ changes their relative phase and their direction along the equator. As the Quantum Atlas puts it, “Qubits along the equator, on the other hand, are equally likely to be found at either pole.”
Four familiar equatorial states
- At φ = 0:
|+x⟩ = (|0⟩ + |1⟩)/√2. - At φ = π:
|−x⟩ = (|0⟩ − |1⟩)/√2. - At φ = π/2:
|+y⟩ = (|0⟩ + i|1⟩)/√2. - At φ = −π/2:
|−y⟩ = (|0⟩ − i|1⟩)/√2.
Each has the same 50–50 probabilities in the z basis, yet the states differ and can produce different outcomes when measured along other axes. In the pure-state description, an equatorial point is a coherent superposition, not a classical mixture of 0 and 1.
Why do the amplitudes use half-angles?
θ is the ordinary polar angle of the Bloch vector, but the state amplitudes use θ/2. This is why the north-pole state has amplitudes 1 and 0 at θ = 0, while at θ = π/2 both amplitudes have magnitude 1/√2. Squaring those magnitudes gives the measurement probabilities. The half-angle relation is a feature of the qubit state parametrization, not a different convention for the sphere’s polar angle; Girvin discusses it in Introduction to Quantum Information.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Are all qubit states on the surface?
No. Pure states lie on the surface, where the Bloch vector has unit length. Mixed states lie inside the sphere, with vector length less than one; the center represents the maximally mixed state. This distinction is covered in the University of Chicago dissertation The Bloch sphere representation of qubit.
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Repair common Windows errors and clear accumulated junk for a smoother, more stable PC - no reinstall needed.Free scan · no reinstallDoes one Bloch sphere represent any quantum system?
No. The ordinary Bloch sphere is a representation for a single qubit. It does not, by itself, represent an arbitrary multi-qubit state. For a single-qubit diagram, also check its angle labels: some diagrams or software may use different spherical-coordinate conventions, so confirm which axis defines θ and where φ begins.
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